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22 HIROAKI NAKAMURA, AKIO TAMAGAWA, SHINICHI MOCHIZUKI give rise to a proof of the sort that will be discussed in §5.2. Concerning the circumstances surrounding this state of affairs, we refer to the discussion of §4.1. 17) In fact, if one takes this as the definition of L, the following argument becomes slightly inaccurate, but in the interest of minimizing the introduction of inessential technical details, we hope that the reader will forgive this minor transgression. 18) <strong>The</strong> phrase “arises from a geometric rational point Spec(L) → X” means that it arises as the morphism Gal(L) =π 1 (Spec(L)) → π 1 (X) → π (p) 1 π 1 to some morphism Spec(L) → X. 19) For more details, we refer to §3.1, (i). 20) For more details, we refer to [M3]. References (X) obtained by applying the functor [AI] G.Anderson, Y.Ihara, Pro-l branched coverings of P 1 and higher circular l-units, Part 1, Ann. of Math. 128 (1988), 271–293; Part 2, Intern. J. Math. 1 (1990), 119–148. [B] G.V.Belyi, On Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk. SSSR 8 (1979), 267–276 (Russian); English transl. in Math. USSR Izv. 14 (1980), no. 2, 247–256. [BK] S.Bloch, K.Kato, L-functions and Tamagawa numbers of motives, <strong>The</strong> <strong>Grothendieck</strong> Festschrift, Volume I, Birkhäuser, 1990, pp. 333–400. [F1] G.Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366. [F2] , p-adic Hodge theory, J. of the Amer. Math. Soc. 1 (1988), 255–299. [SGA1] A.<strong>Grothendieck</strong>, M.Raynaud, Revêtement Etales et Groupe Fondamental (SGA1), Lecture Note in Math., vol. 224, Springer, Berlin Heidelberg New York, 1971. [G1] A.<strong>Grothendieck</strong>, La longue marche à travers de la théorie de Galois, 1981, in preparation by J.Malgoire (first few chapters available since 1996). [G2] , Esquisse d’un Programme, 1984, in [6] vol.1, 7–48. [G3] , Letter to G.Faltings, June 1983, in [6] vol.1, 49–58. [H] D.Harbater, Fundamental groups of curves in characteristic p, Proc.ICM,Zürich (1994), 654–666. [I1] Y.Ihara, Profinite braid groups, Galois representations, and complex multiplications, Ann. of Math. 123 (1986), 43–106. [I2] , Braids, Galois groups and some arithmetic functions, Proc. ICM, Kyoto (1990), 99–120. [IN] Y.Ihara, H.Nakamura, Some illustrative examples for anabelian geometry in high dimensions, in[6] vol.1, 127–138. [MT] M.Matsumoto, A.Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups, Preprint 1997. [M1] S.Mochizuki, <strong>The</strong> profinite <strong>Grothendieck</strong> conjecture for hyperbolic curves over number fields, J. Math. Sci., Univ. Tokyo 3 (1996), 571–627. [M2] , <strong>The</strong> local pro-p <strong>Grothendieck</strong> conjecture for hyperbolic curves, RIMS Preprint 1045, Kyoto Univ. (1995). [M3] , <strong>The</strong> local pro-p anabelian geometry of curves, RIMS Preprint 1097, Kyoto Univ. (1996). [M4] , A <strong>Grothendieck</strong> conjecture-type result for certain hyperbolic surfaces, RIMS Preprint 1104, Kyoto Univ. (1996). [M5] , A theory of ordinary p-adic curves, Publ. of RIMS 32 (1996), 957–1151. [M6] , <strong>The</strong> generalized ordinary moduli of p-adic hyperbolic curves, RIMS Preprint 1051, Kyoto Univ. (1995). [M7] , Combinatorialization of p-adic Teichmüller theory, RIMS Preprint 1076, Kyoto Univ. (1996). [M8] , Correspondences on hyperbolic curves, J. Pure Appl. Algebra 131 (1998), 227–244.
THE GROTHENDIECK CONJECTURE 23 [N1] H.Nakamura, Rigidity of the arithmetic fundamental group of a punctured projective line, J.reine angew. Math. 405 (1990), 117–130. [N2] , Galois rigidity of the étale fundamental groups of punctured projective lines, J. reine angew. Math. 411 (1990), 205–216. [N3] , On galois automorphisms of the fundamental group of the projective line minus three points, Math. Z. 206 (1991), 617–622. [N4] , Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci., Univ. Tokyo 1 (1994), 71–136. [N5] , Galois rigidity of algebraic mappings into some hyperbolic varieties, Intern. J. Math. 4 (1993), 421–438. [N6] , On exterior Galois representations associated with open elliptic curves, J. Math. Sci., Univ. Tokyo 2 (1995), 197–231. [N7] , Fukuyugengun no Garoa Gosei, Sugaku 47 (1995), 1–17; English translation to appear in Sugaku Exposition (AMS). [NTa] H.Nakamura, N.Takao, Galois rigidity of pro-l pure braid groups of algebraic curves, Trans. Amer. Math. Soc. (to appear). [NTs] H.Nakamura, H.Tsunogai, Some finiteness theorems on Galois centralizers in pro-l mapping class groups, J. reine angew. Math. 441 (1993), 115–144. [Ne] J.Neukirch, Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper, Invent. math. 6 (1969), 296–314. [O1] T.Oda, A note on ramification of the Galois representation on the fundamental group of an algebraic curve, J. Number <strong>The</strong>ory 34 (1990), 225–228. [O2] , A note on ramification of the Galois representation on the fundamental group of an algebraic curve, II, J. Number <strong>The</strong>ory 53 (1995), 342–355. [P1] F.Pop, On <strong>Grothendieck</strong>’s conjecture of birational anabelian geometry, Ann. of Math. 138 (1994), 145–182. [P2] , On <strong>Grothendieck</strong>’s conjecture of birational anabelian geometry II, Preprint (June 1995). [T1] A.Tamagawa, <strong>The</strong> <strong>Grothendieck</strong> conjecture for affine curves, Compositio Math. 109 (1997), no. 2, 135–194. [T2] , On the fundamental groups of curves over algebraically closed fields of characteristic > 0, Preprint 1997. [U] K.Uchida, Isomorphisms of Galois groups of algebraic function fields, Ann. of Math. 106 (1977), 589–598. [1] Y.Ihara (ed.), Galois Representations and Arithmetic Algebraic Geometry, Advanced Studies in Pure Math., vol. 12, Kinokuniya Co. Ltd., North-Holland, 1987. [2] Y.Ihara, K.Ribet, J.-P.Serre (eds.), Galois Groups over Q, Math. Sci. Res. Inst. Publications, vol. 16, Springer, 1989. [3] J.-P.Serre, Topics in Galois <strong>The</strong>ory, Jones and Bartlett Publ., 1992. [4] L.Schneps (ed.), <strong>The</strong> <strong>Grothendieck</strong> <strong>The</strong>ory of Dessins d’Enfants, London Math. Soc. Lect. Note Ser., vol. 200, Cambridge Univ. Press, 1994. [5] M.Fried et al. (eds.), Recent Developments in the Inverse Galois Problem, Contemp. Math., vol. 186, AMS, 1995. [6] L.Schneps, P.Lochak (eds.), Geometric Galois Actions; 1.Around <strong>Grothendieck</strong>’s Esquisse d’un Programme, 2.<strong>The</strong> Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, London Math. Soc. Lect. Note Ser., vol. 242-243, Cambridge Univ. Press, 1997.
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